We prove the compatibility of local and global Langlands correspondences for G L n GL_n , which was proved up to semisimplification in M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton-Oxford, 2001. More precisely, for the n n -dimensional l l -adic representation R l ( Π ) R_l(\Pi ) of the Galois group of an imaginary CM-field L L attached to a conjugate self-dual regular algebraic cuspidal automorphic representation Π \Pi of G L n ( A L ) GL_n(\mathbb A_L) , which is square integrable at some finite place, we show that Frobenius semisimplification of the restriction of R l ( Π ) R_l(\Pi ) to the decomposition group of a place v v of L L not dividing l l corresponds to Π v \Pi _v by the local Langlands correspondence. If Π v \Pi _v is square integrable for some finite place v ⧸ | l v \not | l we deduce that R l ( Π ) R_l(\Pi ) is irreducible. We also obtain conditional results in the case v | l v|l .

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We give a purely local proof, in the depth 0 case, of the result by Harris-Taylor which asserts that the local Langlands correspondence for GLn is realized in the vanishing cycle cohomology of the deformation spaces of one-dimensional formal modules of height n. Our proof is given by establishing the direct geometric link with the Deligne-Lusztig theory for GLn(Fq).1 Note added in proof: In the proof of Proposition 6.10, we need a little more argument to prove that U N acts trivially on R j ψΛ. We use Z st to apply Proposition 6.3. Although the first isomorphism of Proposition 6.3 holds only etale locally, it shows that the canonical morphism Λ → R 0 ψΛ| Y 0 J is an isomorphism if d = 1. For general d, by Proposition 6.2, the sheaf R 0 ψΛ is a push forward from a d = 1 situation, namely the normalization of the base change to tamely ramified extension of W of degree d (similar to what is done in §5.1), which restricts to a finite etale covering of degree d on Y 0 J . Thus U N , being a p-group, acts trivially on R 0 ψΛ| Y 0 J . In the second isomorphism of Proposition 6.3, the group U N can only act on the index set J, but J is a partial flag of linear subspaces of P containing N , and U N fixes each element of J.

We give a self-contained exposition of local class field theory, via Lubin-Tate theory and the Hasse-Arf theorem, refining the arguments of Iwasawa [9]. Résumé. Nous présentons une démonstration complète de la théorie du corps de classes locale via la théorie de Lubin-Tate et le théorème de Hasse-Arf, en raffinant des arguments d'Iwasawa [9].